APPLICABLE TO ELLIPTIC AND IJLTEA-ELLIPTIC FUNCTIONS. 
427 
This example has been so chosen as to admit of easy verification. In fact ^=2'30625, 
V “ 4- 1 S 2 Q 
and log ^■ 3 Y=log 2^=0*40330 93959 24. The error is therefore only of three units 
in the tenth decimal place, vehere there was no reason to expect accuracy. 
26. The only other formulae which I shall give are the following, for finding the 
logarithm of a number, and vice versa. They are indispensable where more than seven 
figures are required. 
Let log (.r+A)=logar+^, then 
log +2 — nearly, 
log A=log ^ k nearly. 
The values of log m and logM have been given in paragraph 23. 
27. As an example of the application of the method to the evaluation of elliptic 
integrals of the thh’d class, let us take the integral 
r 
Jg (1 — sin^a.sin^f) (1 — sin^6.sin‘^<p)i 
for the values a=45°, ^=30°, (^ = 60°. 
I have selected these values because they can be obtained without reduction or inter- 
polation from the Table of A(^, ip) which I have given, and also because sin^a=sin^, 
and therefore the integral can be reduced to one of the first class, an inverse tangent, 
thus admitting of easy verification. For this case 
2t= 
(1 — sin^a .sin^ip) ^1 — sin^fl.cos^^.sin^(p^ 
\z^d(p — 
2 sin^a 
-T-. tan ’ (cos a, tan (2)) 
Jtt COSa 
2 sin^flcos^^ 
i 
sin^« — sin®6 . cos® — 
1 — sin® 6 . cos® 
7) ‘‘“"'K 
1— sin® cos® 
7) ’tan?!. 
Making-^ successively 22° 30', 45°, 67° 30', and, for the odd term, 90°, we find, after a 
few obvious reductions, that eight times the value of the integral is 
1 3 +An4^;^+ A»(45°. 22 i) } 45°. tan 60”) 
— 3 qo tan~ ^ { cos 30°. tan 60°} — tan~*{A(30°, 45°). tan 60°} 
_tan-'{A(30°, 67i).tan 60°} .cos® 22i_tan-' |A(30°, 22i) .tan 60°} .cos®67i 
A®(45°,67i).A(30°,67i) A®(45°, 22i).A(30°, 22^) 
3 M 2 
